It is shown first that an algebraic Riccati equation: \(G+A^*X+XA- XFX=0_ n\) may be solved by reducing the Hamiltonian matrix \(H=\left[ \begin{matrix} A^*\quad G\\ F\quad -A\end{matrix} \right]\) to a triangular form: \(Q^*HQ=\left[ \begin{matrix} T^*_ 1\\ 0_ n\end{matrix} \begin{matrix} T_ 2\\ - T_ 1\end{matrix} \right]\) where \(T_ 1\) is lower triangular, \(T^*_ 2=T_ 2\) and Q is unitary and symplectic. The so called QR algorithm [\textit{J. Francis}, Comput. J. 4, 332-345 (1962; Zbl 0104.343)] requiring the construction at each step of a QR (unitary-triangular) factorization is modified such that one takes full advantage of the special structure of the Hamiltonian matrices in the case \(rank(F)=1\). It is shown that the proposed Hamiltonian QR algorithm preserves numerical stability and the Hamiltonian structure requiring significantly less work and storage for problems of size greater than about 20 than the general QR algorithm.